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I believe you left out the effects of a very important
fact here. Protons are *not* elementary particles.
When a proton interacts deeply with another proton asymptotic freedom
becomes relevant. In the collision what may appear to us to be a
proton interacting with another proton is in fact a single quark from
one proton interacting with a single quark of the other proton. The
remaining 4 quarks are oblivious bystanders to what is going on with
the 2 colliding/interacting quarks. This means that all the KE in
these by-standing quarks is essentially *unavailable* for creating
particle-antiparticle pairs.
In any event, the problem has a high cuteness coefficient anyway.